cp's OEIS Frontend

A113079 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (5) in the tree of tournament sequences.

%I A113079 #9 Dec 12 2021 22:55:05
%S A113079 1,5,40,515,10810,376175,22099885,2231417165,393643922005,
%T A113079 123097221805100,69087264010363930,70321483026073531730,
%U A113079 130954011392485408662370,449450774746306949114288795
%N A113079 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (5) in the tree of tournament sequences.
%C A113079 Equals column 5 of square table A093729. Also equals column 0 of the matrix 5th power of triangle A097710, which satisfies the matrix recurrence: A097710(n,k) = [A097710^2](n-1,k-1) + [A097710^2](n-1,k) for n>k>=0.
%H A113079 M. Cook and M. Kleber, <a href="https://doi.org/10.37236/1522">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.
%e A113079 The tree of tournament sequences of descendents of a node labeled (5) begins:
%e A113079 [5]; generation 1: 5->[6,7,8,9,10]; generation 2:
%e A113079 6->[7,8,9,10,11,12], 7->[8,9,10,11,12,13,14],
%e A113079 8->[9,10,11,12,13,14,15,16], 9->[10,11,12,13,14,15,16,17,18],
%e A113079 10->[11,12,13,14,15,16,17,18,19,20]; ...
%e A113079 Then a(n) gives the number of nodes in generation n.
%e A113079 Also, a(n+1) = sum of labels of nodes in generation n.
%o A113079 (PARI) {a(n,q=2)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^5)[n+1,1])}
%Y A113079 Cf. A113077, A113078, A008934, A113089, A113096, A113098, A113100, A113107, A113109, A113111, A113113.
%K A113079 nonn
%O A113079 0,2
%A A113079 _Paul D. Hanna_, Oct 14 2005